$\sqrt{3}$-Based 1-Form Subdivision
Identifieur interne : 000288 ( Main/Corpus ); précédent : 000287; suivant : 000289$\sqrt{3}$-Based 1-Form Subdivision
Auteurs : Jinghao Huang ; Peter SchröderSource :
- Lecture Notes in Computer Science [ 0302-9743 ] ; 2012.
Abstract
Abstract: In this paper we construct an edge based, or 1-form, subdivision scheme consistent with $\sqrt{3}$ subdivision. It produces smooth differential 1-forms in the limit. These can be identified with tangent vector fields, or viewed as edge elements in the sense of finite elements. In this construction, primal (0-form) and dual (2-form) subdivision schemes for surfaces are related through the exterior derivative with an edge (1-form) based subdivision scheme, amounting to a generalization of the well known formulé de commutation. Starting with the classic $\sqrt{3}$ subdivision scheme as a 0-form subdivision scheme, we derive conditions for appropriate 1- and 2-form subdivision schemes without fixing the dual (2-form) subdivision scheme a priori. The resulting degrees of freedom are resolved through spectrum considerations and a conservation condition analogous to the usual moment condition for primal subdivision schemes.
Url:
DOI: 10.1007/978-3-642-27413-8_22
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It produces smooth differential 1-forms in the limit. These can be identified with tangent vector fields, or viewed as edge elements in the sense of finite elements. In this construction, primal (0-form) and dual (2-form) subdivision schemes for surfaces are related through the exterior derivative with an edge (1-form) based subdivision scheme, amounting to a generalization of the well known formulé de commutation. Starting with the classic $\sqrt{3}$ subdivision scheme as a 0-form subdivision scheme, we derive conditions for appropriate 1- and 2-form subdivision schemes without fixing the dual (2-form) subdivision scheme a priori. The resulting degrees of freedom are resolved through spectrum considerations and a conservation condition analogous to the usual moment condition for primal subdivision schemes.</div></front></TEI><istex><corpusName>springer</corpusName><author><json:item><name>Jinghao Huang</name><affiliations><json:string>Applied & Computational Mathematics, Caltech, 1200 E. California Blvd., MC305-16, 91125, Pasadena, CA, USA</json:string><json:string>E-mail: huangjh@caltech.edu</json:string></affiliations></json:item><json:item><name>Peter Schröder</name><affiliations><json:string>Applied & Computational Mathematics, Caltech, 1200 E. 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Starting with the classic $\sqrt{3}$ subdivision scheme as a 0-form subdivision scheme, we derive conditions for appropriate 1- and 2-form subdivision schemes without fixing the dual (2-form) subdivision scheme a priori. The resulting degrees of freedom are resolved through spectrum considerations and a conservation condition analogous to the usual moment condition for primal subdivision schemes.</abstract><qualityIndicators><score>7.606</score><pdfVersion>1.6</pdfVersion><pdfPageSize>429.442 x 659.895 pts</pdfPageSize><refBibsNative>false</refBibsNative><keywordCount>0</keywordCount><abstractCharCount>940</abstractCharCount><pdfWordCount>4486</pdfWordCount><pdfCharCount>23990</pdfCharCount><pdfPageCount>18</pdfPageCount><abstractWordCount>135</abstractWordCount></qualityIndicators><title>$\sqrt{3}$-Based 1-Form Subdivision</title><chapterId><json:string>22</json:string><json:string>Chap22</json:string></chapterId><genre><json:string>conference</json:string></genre><serie><editor><json:item><name>David Hutchison</name><affiliations><json:string>Lancaster University, Lancaster, UK</json:string></affiliations></json:item><json:item><name>Takeo Kanade</name><affiliations><json:string>Carnegie Mellon University, Pittsburgh, PA, USA</json:string></affiliations></json:item><json:item><name>Josef Kittler</name><affiliations><json:string>University of Surrey, Guildford, UK</json:string></affiliations></json:item><json:item><name>Jon M. 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It produces smooth differential 1-forms in the limit. These can be identified with tangent vector fields, or viewed as edge elements in the sense of finite elements. In this construction, primal (0-form) and dual (2-form) subdivision schemes for surfaces are related through the exterior derivative with an edge (1-form) based subdivision scheme, amounting to a generalization of the well known formulé de commutation. Starting with the classic $\sqrt{3}$ subdivision scheme as a 0-form subdivision scheme, we derive conditions for appropriate 1- and 2-form subdivision schemes without fixing the dual (2-form) subdivision scheme a priori. 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University</OrgName><OrgAddress><City>Pittsburgh</City><State>PA</State><Country>USA</Country></OrgAddress></Affiliation><Affiliation ID="Aff3"><OrgName>University of Surrey</OrgName><OrgAddress><City>Guildford</City><Country>UK</Country></OrgAddress></Affiliation><Affiliation ID="Aff4"><OrgName>Cornell University</OrgName><OrgAddress><City>Ithaca</City><State>NY</State><Country>USA</Country></OrgAddress></Affiliation><Affiliation ID="Aff5"><OrgName>ETH Zurich</OrgName><OrgAddress><City>Zurich</City><Country>Switzerland</Country></OrgAddress></Affiliation><Affiliation ID="Aff6"><OrgName>Stanford University</OrgName><OrgAddress><City>Stanford</City><State>CA</State><Country>USA</Country></OrgAddress></Affiliation><Affiliation ID="Aff7"><OrgName>Weizmann Institute of Science</OrgName><OrgAddress><City>Rehovot</City><Country>Israel</Country></OrgAddress></Affiliation><Affiliation ID="Aff8"><OrgName>University of Bern</OrgName><OrgAddress><City>Bern</City><Country>Switzerland</Country></OrgAddress></Affiliation><Affiliation ID="Aff9"><OrgName>Indian Institute of Technology</OrgName><OrgAddress><City>Madras</City><Country>India</Country></OrgAddress></Affiliation><Affiliation ID="Aff10"><OrgName>University of Dortmund</OrgName><OrgAddress><City>Dortmund</City><Country>Germany</Country></OrgAddress></Affiliation><Affiliation ID="Aff11"><OrgName>Massachusetts Institute of Technology</OrgName><OrgAddress><State>MA</State><Country>USA</Country></OrgAddress></Affiliation><Affiliation ID="Aff12"><OrgName>University of California</OrgName><OrgAddress><City>Los Angeles</City><State>CA</State><Country>USA</Country></OrgAddress></Affiliation><Affiliation ID="Aff13"><OrgName>University of California</OrgName><OrgAddress><City>Berkeley</City><State>CA</State><Country>USA</Country></OrgAddress></Affiliation><Affiliation ID="Aff14"><OrgName>Rice University</OrgName><OrgAddress><City>Houston</City><State>TX</State><Country>USA</Country></OrgAddress></Affiliation><Affiliation ID="Aff15"><OrgName>Max-Planck Institute of Computer Science</OrgName><OrgAddress><City>Saarbrücken</City><Country>Germany</Country></OrgAddress></Affiliation></EditorGroup></SeriesHeader><Book Language="En"><BookInfo BookProductType="Proceedings" ContainsESM="No" Language="En" MediaType="eBook" NumberingDepth="2" NumberingStyle="ContentOnly" OutputMedium="All" TocLevels="0"><BookID>978-3-642-27413-8</BookID><BookTitle>Curves and Surfaces</BookTitle><BookSubTitle>7th International Conference, Avignon, France, June 24 - 30, 2010, Revised Selected Papers</BookSubTitle><BookVolumeNumber>6920</BookVolumeNumber><BookSequenceNumber>6920</BookSequenceNumber><BookDOI>10.1007/978-3-642-27413-8</BookDOI><BookTitleID>301227</BookTitleID><BookPrintISBN>978-3-642-27412-1</BookPrintISBN><BookElectronicISBN>978-3-642-27413-8</BookElectronicISBN><BookChapterCount>48</BookChapterCount><BookCopyright><CopyrightHolderName>Springer-Verlag GmbH Berlin Heidelberg</CopyrightHolderName><CopyrightYear>2012</CopyrightYear></BookCopyright><BookSubjectGroup><BookSubject Code="I" Type="Primary">Computer Science</BookSubject><BookSubject Code="I22005" Priority="1" Type="Secondary">Computer Imaging, Vision, Pattern Recognition and Graphics</BookSubject><BookSubject Code="I22013" Priority="2" Type="Secondary">Computer Graphics</BookSubject><BookSubject Code="I21025" Priority="3" Type="Secondary">Simulation and Modeling</BookSubject><BookSubject Code="I23044" Priority="4" Type="Secondary">Computer-Aided Engineering (CAD, CAE) and Design</BookSubject><BookSubject Code="I22021" Priority="5" Type="Secondary">Image Processing and Computer Vision</BookSubject><BookSubject Code="I17028" Priority="6" Type="Secondary">Discrete Mathematics in Computer Science</BookSubject><SubjectCollection Code="SUCO11645">Computer Science</SubjectCollection></BookSubjectGroup><BookContext><SeriesID>558</SeriesID></BookContext></BookInfo><BookHeader><EditorGroup><Editor AffiliationIDS="Aff16"><EditorName DisplayOrder="Western"><GivenName>Jean-Daniel</GivenName><FamilyName>Boissonnat</FamilyName></EditorName><Contact><Email>jean-daniel.boissonnat@sophia.inria.fr</Email></Contact></Editor><Editor AffiliationIDS="Aff17"><EditorName DisplayOrder="Western"><GivenName>Patrick</GivenName><FamilyName>Chenin</FamilyName></EditorName><Contact><Email>patrick.chenin@imag.fr</Email></Contact></Editor><Editor AffiliationIDS="Aff18"><EditorName DisplayOrder="Western"><GivenName>Albert</GivenName><FamilyName>Cohen</FamilyName></EditorName><Contact><Email>cohen@ann.jussieu.fr</Email></Contact></Editor><Editor AffiliationIDS="Aff19"><EditorName DisplayOrder="Western"><GivenName>Christian</GivenName><FamilyName>Gout</FamilyName></EditorName><Contact><Email>christian.gout@insa-rouen.fr</Email></Contact></Editor><Editor AffiliationIDS="Aff20"><EditorName DisplayOrder="Western"><GivenName>Tom</GivenName><FamilyName>Lyche</FamilyName></EditorName><Contact><Email>tom@ifi.uio.no</Email></Contact></Editor><Editor AffiliationIDS="Aff21"><EditorName DisplayOrder="Western"><GivenName>Marie-Laurence</GivenName><FamilyName>Mazure</FamilyName></EditorName><Contact><Email>marie-laurence.mazure@imag.fr</Email></Contact></Editor><Editor AffiliationIDS="Aff22"><EditorName DisplayOrder="Western"><GivenName>Larry</GivenName><FamilyName>Schumaker</FamilyName></EditorName><Contact><Email>larry.schumaker@vanderbilt.edu</Email></Contact></Editor><Affiliation ID="Aff16"><OrgName>INRIA Sophia-Antipolis</OrgName><OrgAddress><Street>2004 Route des Lucioles</Street><Postbox>BP 93</Postbox><Postcode>06902</Postcode><City>Sophia Antipolis</City><Country>France</Country></OrgAddress></Affiliation><Affiliation ID="Aff17"><OrgDivision>Laboratoire LJK</OrgDivision><OrgName>University Joseph Fourier</OrgName><OrgAddress><Postbox>BP 53</Postbox><Postcode>38420</Postcode><City>Grenoble Cedex 9</City><Country>France</Country></OrgAddress></Affiliation><Affiliation ID="Aff18"><OrgDivision>Laboratoire Jacques-Louis Lions</OrgDivision><OrgName>Université Pierre et Marie Curie</OrgName><OrgAddress><Street>4, Place Jussieu</Street><Postcode>75005</Postcode><City>Paris</City><Country>France</Country></OrgAddress></Affiliation><Affiliation ID="Aff19"><OrgDivision>Laboratoire de Mathématiques</OrgDivision><OrgName>INSA Rouen</OrgName><OrgAddress><Street>Avenue de l’Université</Street><Postcode>76800</Postcode><City>St. Etienne du Rouvray</City><Country>France</Country></OrgAddress></Affiliation><Affiliation ID="Aff20"><OrgName>University of Oslo, DMA</OrgName><OrgAddress><Street>Blindern</Street><Postbox>P.O.Box 1053</Postbox><Postcode>0316</Postcode><City>Oslo</City><Country>Norway</Country></OrgAddress></Affiliation><Affiliation ID="Aff21"><OrgDivision>Laboratoire LJK</OrgDivision><OrgName>Université Joseph Fourier</OrgName><OrgAddress><Postbox>BP 53</Postbox><Postcode>38041</Postcode><City>Grenoble Cedex 9</City><Country>France</Country></OrgAddress></Affiliation><Affiliation ID="Aff22"><OrgDivision>Department of Mathematics</OrgDivision><OrgName>Vanderbilt University</OrgName><OrgAddress><Postcode>37240</Postcode><City>Nashville</City><State>TN</State><Country>USA</Country></OrgAddress></Affiliation></EditorGroup></BookHeader><Chapter ID="Chap22" Language="En"><ChapterInfo ChapterType="OriginalPaper" ContainsESM="No" NumberingDepth="2" NumberingStyle="ContentOnly" TocLevels="0"><ChapterID>22</ChapterID><ChapterDOI>10.1007/978-3-642-27413-8_22</ChapterDOI><ChapterSequenceNumber>22</ChapterSequenceNumber><ChapterTitle Language="En"><InlineEquation ID="IEq1"><InlineMediaObject><ImageObject FileRef="978-3-642-27413-8_22_Chapter_TeX2GIF_IEq1.gif" Format="GIF" Color="BlackWhite" Type="Linedraw" Rendition="HTML"></ImageObject></InlineMediaObject><EquationSource Format="TEX">$\sqrt{3}$</EquationSource></InlineEquation>-Based 1-Form Subdivision</ChapterTitle><ChapterFirstPage>351</ChapterFirstPage><ChapterLastPage>368</ChapterLastPage><ChapterCopyright><CopyrightHolderName>Springer-Verlag Berlin Heidelberg</CopyrightHolderName><CopyrightYear>2012</CopyrightYear></ChapterCopyright><ChapterGrants Type="Regular"><MetadataGrant Grant="OpenAccess"></MetadataGrant><AbstractGrant Grant="OpenAccess"></AbstractGrant><BodyPDFGrant Grant="Restricted"></BodyPDFGrant><BodyHTMLGrant Grant="Restricted"></BodyHTMLGrant><BibliographyGrant Grant="Restricted"></BibliographyGrant><ESMGrant Grant="Restricted"></ESMGrant></ChapterGrants><ChapterContext><SeriesID>558</SeriesID><BookID>978-3-642-27413-8</BookID><BookTitle>Curves and Surfaces</BookTitle></ChapterContext></ChapterInfo><ChapterHeader><AuthorGroup><Author AffiliationIDS="Aff23"><AuthorName DisplayOrder="Western"><GivenName>Jinghao</GivenName><FamilyName>Huang</FamilyName></AuthorName><Contact><Email>huangjh@caltech.edu</Email></Contact></Author><Author AffiliationIDS="Aff23"><AuthorName DisplayOrder="Western"><GivenName>Peter</GivenName><FamilyName>Schröder</FamilyName></AuthorName><Contact><Email>huangjh@caltech.edu</Email></Contact></Author><Affiliation ID="Aff23"><OrgName>Applied & Computational Mathematics, Caltech</OrgName><OrgAddress><Street>1200 E. California Blvd., MC305-16</Street><City>Pasadena</City><State>CA</State><Postcode>91125</Postcode><Country>USA</Country></OrgAddress></Affiliation></AuthorGroup><Abstract ID="Abs1" Language="En"><Heading>Abstract</Heading><Para>In this paper we construct an edge based, or <Emphasis Type="Italic">1-form</Emphasis>, subdivision scheme consistent with <InlineEquation ID="IEq2"><InlineMediaObject><ImageObject FileRef="978-3-642-27413-8_22_Chapter_TeX2GIF_IEq2.gif" Format="GIF" Color="BlackWhite" Type="Linedraw" Rendition="HTML"></ImageObject></InlineMediaObject><EquationSource Format="TEX">$\sqrt{3}$</EquationSource></InlineEquation> subdivision. It produces smooth differential 1-forms in the limit. These can be identified with tangent vector fields, or viewed as edge elements in the sense of finite elements. In this construction, primal (0-form) and dual (2-form) subdivision schemes for surfaces are related through the exterior derivative with an edge (1-form) based subdivision scheme, amounting to a generalization of the well known formulé de commutation.</Para><Para>Starting with the classic <InlineEquation ID="IEq3"><InlineMediaObject><ImageObject FileRef="978-3-642-27413-8_22_Chapter_TeX2GIF_IEq3.gif" Format="GIF" Color="BlackWhite" Type="Linedraw" Rendition="HTML"></ImageObject></InlineMediaObject><EquationSource Format="TEX">$\sqrt{3}$</EquationSource></InlineEquation> subdivision scheme as a 0-form subdivision scheme, we derive conditions for appropriate 1- and 2-form subdivision schemes without fixing the dual (2-form) subdivision scheme a priori. The resulting degrees of freedom are resolved through spectrum considerations and a conservation condition analogous to the usual moment condition for primal subdivision schemes.</Para></Abstract><KeywordGroup Language="En"><Heading>Keywords</Heading><Keyword>Discrete exterior calculus</Keyword><Keyword><InlineEquation ID="IEq4"><InlineMediaObject><ImageObject FileRef="978-3-642-27413-8_22_Chapter_TeX2GIF_IEq4.gif" Format="GIF" Color="BlackWhite" Type="Linedraw" Rendition="HTML"></ImageObject></InlineMediaObject><EquationSource Format="TEX">$\sqrt3$</EquationSource></InlineEquation>-subdivision</Keyword><Keyword>1- and 2-form subdivision</Keyword><Keyword>tangent vector field generation</Keyword></KeywordGroup></ChapterHeader><NoBody></NoBody></Chapter></Book></Series></Publisher></istex:document></istex:metadataXml><mods version="3.6"><titleInfo lang="en"><title>$\sqrt{3}$-Based 1-Form Subdivision</title></titleInfo><titleInfo type="alternative" contentType="CDATA" lang="en"><title>$\sqrt{3}$ -Based 1-Form Subdivision</title></titleInfo><name type="personal"><namePart type="given">Jinghao</namePart><namePart type="family">Huang</namePart><affiliation>Applied & Computational Mathematics, Caltech, 1200 E. California Blvd., MC305-16, 91125, Pasadena, CA, USA</affiliation><affiliation>E-mail: huangjh@caltech.edu</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">Peter</namePart><namePart type="family">Schröder</namePart><affiliation>Applied & Computational Mathematics, Caltech, 1200 E. California Blvd., MC305-16, 91125, Pasadena, CA, USA</affiliation><affiliation>E-mail: huangjh@caltech.edu</affiliation><role><roleTerm type="text">author</roleTerm></role></name><typeOfResource>text</typeOfResource><genre type="conference" displayLabel="OriginalPaper"></genre><originInfo><publisher>Springer Berlin Heidelberg</publisher><place><placeTerm type="text">Berlin, Heidelberg</placeTerm></place><dateIssued encoding="w3cdtf">2012</dateIssued><copyrightDate encoding="w3cdtf">2012</copyrightDate></originInfo><language><languageTerm type="code" authority="rfc3066">en</languageTerm><languageTerm type="code" authority="iso639-2b">eng</languageTerm></language><physicalDescription><internetMediaType>text/html</internetMediaType></physicalDescription><abstract lang="en">Abstract: In this paper we construct an edge based, or 1-form, subdivision scheme consistent with $\sqrt{3}$ subdivision. It produces smooth differential 1-forms in the limit. These can be identified with tangent vector fields, or viewed as edge elements in the sense of finite elements. In this construction, primal (0-form) and dual (2-form) subdivision schemes for surfaces are related through the exterior derivative with an edge (1-form) based subdivision scheme, amounting to a generalization of the well known formulé de commutation. Starting with the classic $\sqrt{3}$ subdivision scheme as a 0-form subdivision scheme, we derive conditions for appropriate 1- and 2-form subdivision schemes without fixing the dual (2-form) subdivision scheme a priori. The resulting degrees of freedom are resolved through spectrum considerations and a conservation condition analogous to the usual moment condition for primal subdivision schemes.</abstract><relatedItem type="host"><titleInfo><title>Curves and Surfaces</title><subTitle>7th International Conference, Avignon, France, June 24 - 30, 2010, Revised Selected Papers</subTitle></titleInfo><name type="personal"><namePart type="given">Jean-Daniel</namePart><namePart type="family">Boissonnat</namePart><affiliation>INRIA Sophia-Antipolis, 2004 Route des Lucioles, BP 93, 06902, Sophia Antipolis, France</affiliation><affiliation>E-mail: jean-daniel.boissonnat@sophia.inria.fr</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Patrick</namePart><namePart type="family">Chenin</namePart><affiliation>Laboratoire LJK, University Joseph Fourier, BP 53, 38420, Grenoble Cedex 9, France</affiliation><affiliation>E-mail: patrick.chenin@imag.fr</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Albert</namePart><namePart type="family">Cohen</namePart><affiliation>Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 4, Place Jussieu, 75005, Paris, France</affiliation><affiliation>E-mail: cohen@ann.jussieu.fr</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Christian</namePart><namePart type="family">Gout</namePart><affiliation>Laboratoire de Mathématiques, INSA Rouen, Avenue de l’Université, 76800, St. Etienne du Rouvray, France</affiliation><affiliation>E-mail: christian.gout@insa-rouen.fr</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Tom</namePart><namePart type="family">Lyche</namePart><affiliation>University of Oslo, DMA, Blindern, P.O.Box 1053, 0316, Oslo, Norway</affiliation><affiliation>E-mail: tom@ifi.uio.no</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Marie-Laurence</namePart><namePart type="family">Mazure</namePart><affiliation>Laboratoire LJK, Université Joseph Fourier, BP 53, 38041, Grenoble Cedex 9, France</affiliation><affiliation>E-mail: marie-laurence.mazure@imag.fr</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Larry</namePart><namePart type="family">Schumaker</namePart><affiliation>Department of Mathematics, Vanderbilt University, 37240, Nashville, TN, USA</affiliation><affiliation>E-mail: larry.schumaker@vanderbilt.edu</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><genre type="book-series" displayLabel="Proceedings"></genre><originInfo><copyrightDate 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type="family">Hutchison</namePart><affiliation>Lancaster University, Lancaster, UK</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Takeo</namePart><namePart type="family">Kanade</namePart><affiliation>Carnegie Mellon University, Pittsburgh, PA, USA</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Josef</namePart><namePart type="family">Kittler</namePart><affiliation>University of Surrey, Guildford, UK</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Jon</namePart><namePart type="given">M.</namePart><namePart type="family">Kleinberg</namePart><affiliation>Cornell University, Ithaca, NY, USA</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Friedemann</namePart><namePart type="family">Mattern</namePart><affiliation>ETH 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